The Mirror say the Foxes ace is ‘closing in’ on a dream move to Stamford Bridge.
Chelsea have doubled their opening offer of £15m for the England international and are much closer to the 2016 Premier League champions’ £40m valuation.
Drinkwater is desperate to make the move and Chelsea are hoping to strike a deal as Leicester know he wants out.
Antonio Conte is working hard to bring in reinforcements after losing a number of players this summer.
Four have been signed but he confirmed at the weekend that efforts were still being made behind the scenes to expand his squad.
And the Mirror say Drinkwater is the man most likely to be arriving before the window closes in August 31.
Foxes boss Craig Shakespeare has insisted he will carry on as usual despite interest in Drinkwater and Riyad Mahrez.
He said: “I’ve read the speculation the same as you but until we get the bids accepted, there’s no difference for me.
“We carry on with them. All the time we’re talking about the speculation. We’ll carry on as normal until I’m told otherwise.
“I’ve said that your worst fear is that an acceptable bid comes in, but if it does, we have to deal with it. Until it does, we have no decision to make.”
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If Vardy does arrive, it could be bad news for back up option Michy Batshuayi. The Mirror believe that French club Lille could come in with a £36 million offer for the striker which would allow him to return to regular first team football and to France. That would represent a £3 million profit in 12 months for the Blues if accepted.
However, the Daily Mail claim that the Belgian will turn down interest from both Lille and West Ham in order to stay at Stamford Bridge and fight for his place. He feels confident that he has proven himself enough to have a shot at regular football at Chelsea this season after an impressive pre-season.
The Daily Mail also report that Chelsea and Leicester City have reached a standstill in negotiations over midfielder Danny Drinkwater. The former Manchester United youth product is keen on the switch but Chelsea are refusing to raise their £30 million valuation to meet Leicester’s, which currently stands at £40 million.
Craig Shakespeare’s side have shown that they will not be forced to sell any player against their will as they have held on to winger Riyad Mahrez despite a transfer request at the start of the summer. Drinkwater may be keen on the move but Chelsea will have to pay up if they are to get their man.